Question

An A.P. with 3^rd term = −8 and 9^th term = 4. 

Assertion (A): Common difference, d = −2. 

Reason (R): If first term of the A.P. is a, then (a + 8d) − (a + 2d) = 4 + 8.

विकल्प

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

A is false, R is true.

Explanation:

Given, in an A.P. 3^rd term = −8 and 9^th term = 4.

Let a be the first term of the A.P. and d be the common difference.

By formula :

⇒ a_n = a + (n − 1)d

⇒ a₃ = a + (3 − 1)d

⇒ −8 = a + 2d  ..........(1)

⇒ a₉ = a + (9 − 1)d

⇒ 4 = a + 8d  ..........(2)

Subtracting equation (1) from (2), we get :

⇒ (a + 8d) − (a + 2d) = 4 − (−8)

⇒ a + 8d − a − 2d = 4 + 8

⇒ 6d = 12

⇒ d = `12/6`

⇒ d = 2.

According to assertion d = −2, which is incorrect.

So, assertion (A) is false.

From above calculation, we get :

⇒ (a + 8d) − (a + 2d) = 4 − (−8)

⇒ (a + 8d) − (a + 2d) = 12

According to reason,

⇒ (a + 8d) − (a + 2d) = −8 − 4

⇒ (a + 8d) − (a + 2d) = −12, which is incorrect.

So, reason (R) is false.